In [1]}, it was shown that if a spacetime \((M,g)\)  is globally hyperbolic and future timelike geodesically complete, then \(\partial ^+M = \emptyset \) , and in [2]}, it was shown that the assumption of global hyperbolicity is superfluous. Hence future timelike completeness implies that the past boundary \(\partial ^-M\)  is an achronal topological hypersurface by Lemma REF . Combining this with (d\(^{\prime }\) ), we see that \(\partial ^-M\)  looks like figure REF , at least locally near \(\mathcal {O}\) .
